How Do You Calculate Force and Tension in a Pulley System?

[Apprentice123]

The weights of the blocks A, B and C, shown in figure worth, 89N, 44,5 N and 44,5 N, respectively. The blocks are initially at rest. Knowing that B travels 2,44 m in 2s, determine (a) the module of F and (b) the tension on the rope AD. Discard the masses of the pulleys and friction.

Answer:
(a) 7,37 N
(b) 92,7 N

My solution

Block B
ZFy = mB . aB
T - F - mB.g = mB . aB

Block C
ZFy = mC . aC
-mC . g + T = mC . aC

aB = -aC

I can not find the answer of the book

 

[rl.bhat]

Using the formula s = 1/2*at^2, find the acceleration of mb.

 

[tiny-tim]

Hi Apprentice123!

aB = -aC

Nooo … not this time! …

because … ? ​

 

[Apprentice123]

Hi Apprentice123!


Nooo … not this time! …

because … ? ​

Because ? I do not know
I have:

x = 1/2 * aB * t^2
aB = 1.22 m/s^2


Block B
T - F - PB = mB * aB
F = T - 50,034

Block C
-PC + T = mC * aC
T = 44,5 + 4,5361aC

How to calculate aC?

 

[rl.bhat]

For block B, it should be
F + mB*g - T = mB*aB

 

[Apprentice123]

For block B, it should be
F + mB*g - T = mB*aB

Yes, thanks. How do I calculate aC?

 

[tiny-tim]

Hi Apprentice123!

Yes, thanks. How do I calculate aC?

General tip: if you can't see clearly what to do, then give everything names (or letters) …

call the lengths of the ropes above A B C and D a b c and d, respectivley.

Then the height of B is -(b + d), also b + c is constant, and so is a + d.

Carry on from there …

 

[andrevdh]

What is this module, first time for me? Don't you think it is the resultant force acting on B?

 

[rl.bhat]

What is this module, first time for me? Don't you think it is the resultant force acting on B?

No. If you observe the weights of A, B and C , You can see that they are in equilibrium. To start the motion, a force F is applied on B.

 

[andrevdh]

So the accelerations of B and C will differ due to the fact that the acceleration of pulley D "adds to" the motion of B and "subtracts from" the motion C?

 

[leena19]

I have a very similar question that I'm having problems solving,
For mass B,
$$\downarrow$$F + 44.5 - T = 4.45a
and a=1.22ms^-2

For C,
$$\uparrow$$ T-44.5 = ma'

At first I thought a=a' and I tried working with it but I get a different answer,and after reading the other posts,I'm not sure they(the accelerations) are the same,but I don't understand why they are not either and how I'm supposed to calculate it.

Hope my problem's clear and I hope someone can help me
Thanks in advance

 

[tiny-tim]

I have a very similar question …
…
At first I thought a=a' and I tried working with it but I get a different answer,and after reading the other posts,I'm not sure they(the accelerations) are the same,but I don't understand why they are not either and how I'm supposed to calculate it.

Hi leena19!

See post #7 …

The height of B is -(b + d), and so the accleration of B is -(b'' + d''), while for C it's … ?

 

[leena19]

Hello tiny-tim

See post #7 …

I've seen it,sadly I don't understand it :(

I understand ,why b+c and c+d is constant.Also,why a+b+d and a+c+d is constant,cause the length of the string's constant.
but i don't get why the height of B is negative(b+d),are we taking displacements here?

The height of B is -(b + d), and so the accleration of B is -(b'' + d''), while for C it's … ?

i don't know.is it -(c+d) or (c+d) ? and you get aB = -(b'' + d'') ,by double differentiaion,is it?
When i use,
a+b+d , i get aA + aB = 0,and aA = -aB
and for a+c+d,i get aA = -aC ,which can't be right,right?
I've never dealt with 2 pulley problems before,but I've done a lot of single pulley problems,but we've never had to consider the heights there to find the acceleration,so I'm a bit lost here.

THANK YOU

 

[leena19]

This is an old thread I found on double pulleys

https://www.physicsforums.com/archive/index.php/t-258463.html"

I understand here why the acceleration is doubled for m1,as explained in the thread,but
I don't get why we can't apply the same principle here.
I also can't understand how the accelerations of B and C are different,in this system of 3 pulleys?and what the heights of the weights' got to do with it?
I think the acceleration of A is 1/2 that of B &/or 1/2 the acceleration of C?

I hope someone can explain to me the motion of this pulley system.

Thank you

 

[andrevdh]

Accelerations are vectors, and since the motion here is one dimensional one can add/subtract the various accelerations that the components in this system is experiencing. This also follows from Newton's second law, that is each force in the resultant equation for each mass has its own contribution to the resultant acceleration.

That being said one can see that D and A will share the same acceleration. D's acceleration is downwards in this situation. B and C would have had the same acceleration if it were not for the acceleration of D.

B accelerates downwards and C upwards. This mean that their common acceleration will be altered as follows:

The acceleration of D will subtract from their common acceleration for C and add for the acceleration of B.

 

[anant25121986]

Hi all,

I tried solving this. And got a different answer. Below is what I did.

Let us assume that acceleration of a block is positive in downward direction (as B moves down). Let the tension in the rope A-D be T1 and that in rope BC be T2.

Note that the system is an accelerating system. This implies that pulley D is going down with acceleration aA. Also with respect to D, accelerations of B and C need to be equal but in opposite directions. If accelerations of B and C wrt to D are aBD and aCD respectively, then

aB=aA+aBD aB is acceleration of B with respect to fixed point of pulleys between A and D
aC=aA+aCD aC is acceleration of C with respect to fixed point of pulleys between A and D

aCD=-aBD
Hence, aC=aA-aBD

For block B,
F+mB*g-T2=mB*aBD

For block C,
mC*g-T2=mC*aCD

For pulley D (as its massless),
2*T2=T1

For block A,
T1-mA*g = mA*aA

Now, we have been given the values of the weights in N (i.e. in the form of mX*g). Also given, is the displacement of B and the duration for the displacement.

aB=2*s*t^2 gives us aB=1.22 m/s^2

assuming g=10 m/s^2 gives us mA=8.9 kg, mB=mC=4.45 kg, we get for block B,

From here on, mA=2m and mB=mC=m. Adding equations obtained for block B and block C,
F + 2*m*g - 2*T2 = 0 as aCD = - aBD
F + 2*m*g = T1 ------------- (1)

Subtracting them gives,
F = 2*m*aBD ------------- (2)

From equation for block A we have,
T1 - 2*m*g = 2*m*aA
T1 = 2*m*aA + 2*m*g -------------- (3)

Using (3) in (1),
F = 2*m*aA

Compare this (2),
2*m*aA = 2*m*aBD
aBD = aA (:D moment)

As aB = aA + aBD,
aA = 0.5 * aB = 0.61 m/s^2

F = 2*m*aA = 54.29 N
T1 = F1 + 2*m*g = 143.29 N

If someone can find my mistake kindly assist. Already grateful to those who can help.

 

[anant25121986]

F = 2*m*aA = 54.29 N
T1 = F1 + 2*m*g = 143.29 N


I made a mistake here. The values I got were,

F = 5.429 N
T1 = 5.429 + 89 = 94.429 N

Still very different from the answer. :(

 

[leena19]

When you apply a force F on B, it moves down a distance b. It pulls A and C up with distance a and c respectively.
So b should be equal to a + c

Thank you rl.bhat.This is all that I needed to know.
But just to makesure I've understood it fully, if we were to take the distance interms of d when B is moving downwards,would it be like this?
b-c+d = 0
b = c-d
since -d = +a,
b = c+a, or
since aB = aC -AD ,and aD =-aA
aB = aC + aA

 

[andrevdh]

anant,

...

For block B,
F+mB*g-T2=mB*aBD

For block C,
mC*g-T2=mC*aCD

...

you should use the accelerations aB and aC here, that is the forces are such so that the masses obtained these resultant accelerations - Newton's second law.

 

[anant25121986]

you should use the accelerations aB and aC here, that is the forces are such so that the masses obtained these resultant accelerations - Newton's second law.

Still not with you. Could you please explain?

 

[rl.bhat]

Thank you rl.bhat.This is all that I needed to know.
But just to makesure I've understood it fully, if we were to take the distance interms of d when B is moving downwards,would it be like this?
b-c+d = 0
b = c-d
since -d = +a,
b = c+a, or
since aB = aC -AD ,and aD =-aA
aB = aC + aA

I have observed a flaw in my post #18. Here is a new approach.
The given system contains two ropes. Each rope has two segments. when the force F is applied, all the segments will change their lengths. Note down the changes in the segments of a rope,and equate the net change to zero. Here we go.
When a force F is applied on B, the displacements of different masses and pulley D are:
B down by b, C up by c, D down by d and A up by a.
Now let us consider change in length of rope connecting Band C.
Since D moves down by d, both B and C will move down by d. Hence
net change in segment B = b-d
net change in segment C = -c-d
So the net change in the rope = b-d-c-d = 0. Or 2d = b - c.
And in rope connecting A and D, a = d.
Hence 2aA = aB - aC. Or aB = 2aA + aC
(F + mB*g - T)/mB = 2( 2T - mA*g)/mA + ( T - mC*g)
1.22 = 2(2T - 89)/8.9 + ( T - 44.5 )/4.45
= (6T - 267)/8.9
1.22*8.9 = 6T - 267
6T = 277.9
Tension in AD = 2T = 92.62 N.

 

[leena19]

The given system contains two ropes. Each rope has two segments. when the force F is applied, all the segments will change their lengths. Note down the changes in the segments of a rope,and equate the net change to zero. Here we go.
When a force F is applied on B, the displacements of different masses and pulley D are:
B down by b, C up by c, D down by d and A up by a.
Now let us consider change in length of rope connecting Band C.
Since D moves down by d, both B and C will move down by d.

I understand all of the above,rl.bhat.
It is this part that I find difficult to understand,

net change in segment B = b-d
net change in segment C = -c-d

We're taking displacements here,right?I think that's what you mean by net change in segment?so
When I, tried taking displacements,I end up with,
$$\uparrow$$ B= -b +d
C = c-d
net change in rope = -b - d + c - d = 0
and 2d = c - b ,which I know is wrong ,cause when I tried do it like this,I don't get the answer,but I really don't know why it's wrong? :(

 

[rl.bhat]

I understand all of the above,rl.bhat.
It is this part that I find difficult to understand,

We're taking displacements here,right?I think that's what you mean by net change in segment?so
When I, tried taking displacements,I end up with,
$$\uparrow$$ B= -b +d
C = c-d
net change in rope = -b - d + c - d = 0
and 2d = c - b ,which I know is wrong ,cause when I tried do it like this,I don't get the answer,but I really don't know why it's wrong? :(

You are applying the force to B. So the length of the segment B is increasing. There fore b should be positive. Similarly C is going up. So the length of the segment C is decreasing. Therefore c should be negative. Change of the length of the segment is measured as the final length - initial length.

 

[andrevdh]

anant,

Your accelerations aBD and aCD is the accelerations of B and C w.r.t. D, and since D is also accelerating these are not the resultant accelerations. Newton's second law states that the sum of the forces acting on a body is equal to the resultant acceleration times its mass, which is your accelerations aB and aC.

Also the accelerations aBD and aCD will be such that

aBD = - aCD

which means that your statements:

" aB=aA+aBD aB is acceleration of B with respect to fixed point of pulleys between A and D
aC=aA+aCD aC is acceleration of C with respect to fixed point of pulleys between A and D"

should rather be

aB=aA+aBD
aC=aA-aBD

 

[leena19]

You are applying the force to B. So the length of the segment B is increasing. There fore b should be positive. Similarly C is going up. So the length of the segment C is decreasing. Therefore c should be negative. Change of the length of the segment is measured as the final length - initial length.

I'm sorry,rl.bhat.I still don't get it.
for B
final length= b+d
initial length = b
so substracting gives me -b

and for C,
final length = d-c
initial length = c
and subtacting to find the net change in segment gives me = d-2c ?

 

[rl.bhat]

I'm sorry,rl.bhat.I still don't get it.
for B
final length= b+d
initial length = b
so substracting gives me -b

and for C,
final length = d-c
initial length = c
and subtacting to find the net change in segment gives me = d-2c ?

In the problem it is said that B moves 2.44 m in 2 seconds.
In 2.44 m(b) includes fall of the mass B, which changes the length of the rope segment, plus fall of the pulley D(d), which does not change the length of the rope segment
To calculate the net change in the length of the rope, we must know the change in the rope segment of the mass B alone.And that is b - d

 

[leena19]

To calculate the net change in the length of the rope, we must know the change in the rope segment of the mass B alone.And that is b - d

OK.Thanks.
I think I understand now,but I would like to know if this is how we generally solve for the accelerations in all pulley systems,that is by taking the displacements and then the change in rope segments?

 

[leena19]

Forget my last post.I still don't get it

Since the movement of pulley D has no effect on the rope segment of B,shouldn't the change in rope segment of B,be b only?

 

[tiny-tim]

Forget my last post.I still don't get it

Since the movement of pulley D has no effect on the rope segment of B,shouldn't the change in rope segment of B,be b only?

Hi leena19!

I haven't really been following this thread, but the depth of B (below the ceiling) is Lb + Ld, the depth of C is Lc + Ld, the depth of D is Ld, the depth of A is La, and La + Ld is constant, and Lb + Lc is constant …

you should be able to work it all out from that.

(Lb is the length of rope between B and the nearest pulley, and so on )

 

[rl.bhat]

Forget my last post.I still don't get it

Since the movement of pulley D has no effect on the rope segment of B,shouldn't the change in rope segment of B,be b only?

Even though the movement of pulley D has no effect on the rope segment of B, it affects the distance moved by the body. In the problem b is not the change in length of the rope segment. It is the distance moved by B. If D is not moving, then b is the change in length of the rope segment B.
In all such problems we have to follow five steps.
Step 1: Find out the number of inextensible ropes. For every inextensible rope there will be one equation.
Step 2: For every inextensible rope, find the bodies that are connected to the rope.
Step 3: To relate the acceleration between the bodies, assume that the various bodies move by a distance x1, x2, ...and so on.Calculate the number of segments in the rope.
Step 4: Relate the distance moved. (Total change in the length of the rope must be zero.) To do these calculate the change in length of each segment of the rope. Then add these changes to get the total change in the rope.
Step 5: Once we get the relation between the distances moved, the acceleration relation will be the same.

 

[leena19]

Thank you SO much,rl.bhat.
I finally understand.It feels so good to finally understand,
so once again ithank you rl.bhat,tiny-tim,andrevdh and anant for all the help!

 

[rl.bhat]

Thank you SO much,rl.bhat.
I finally understand.It feels so good to finally understand,
so once again ithank you rl.bhat,tiny-tim,andrevdh and anant for all the help!

You are welcome. I am happy that finally I am able to convince you.